Graphics Reference
In-Depth Information
Figure 12.4.
Interesting surfaces of revolution.
Since p(1,p/4) = p , it follows that
p
x
p
) = (
)
(
)
(
(
) =-
14 11212
,
p
,
,
and
14 0 22
,
p
,
,
∂q
are a basis for our tangent plane. Finally,
p
∂q
p
(
)
(
) ¥
(
) =- -
14
,
p
14 2 2 2
,
p
,
,
x
is a normal vector to the plane, so that its equation is
(
) (
(
) - (
)
) =
22 2
,
--
,
xyz
, ,
122 0
,
,
,
or
22220
x
-
y
-
z
+
=
.
12.2.2 Example. Assume that S is the surface of revolution obtained by rotating
the segment X = [(3,1),(3,3)] about the x-axis. We want to find the tangent plane to S
at p = (3,0,2).
Solution.
The surface S is parameterized by
(
) = (
)
py
,
q
3
,
y
cos ,
q
y
sin
q
and
p
y
∂q
p
(
) = (
)
(
) =-
(
)
y
,
q
0
,cos ,sin
q
q
and
y
,
q
0
,
y
sin ,
q
y
cos
q
.
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